Tagged Questions

A vector space $\mathfrak{g}$ over some field $F$ and kitted with a bilinear, antisymmetric and Jacobi-identity-fulfilling product ("Lie Bracket" or "commutator"). In physics, most often arises as the Lie algebra (tangent space to the identity) of a Lie group; in gauge theories, basis vectors of the ...

Let's say that I have finite chiral transform and I would like to show invariance of Dirac's Lagrangian when $m=0$ under it.
The chiral transform is defined as:
$$\psi(x) \rightarrow \psi'(x) =e^{i ...

I have been look all across the internet and every book I could find trying to get a full derivation of the generator of rotations and more specifically angular momentum as a generator of rotations. I ...

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...

What is meant by su(2) level k algebra ? Is it a lie algebra of some lie group ? What is the relation with SU(2) group. I see it in the context of quantum hall edges.
Googling and google-booking for a ...

In D=2, we can have locally analytic transformations that cannot be globally well-defined.
However, for CFTs in D>2, we have only the global group. Why is that?
Also, is it a statement that depends ...

It is common knowledge that a composition of boosts is not a boost, but involves a rotation. Further, in discussions of Thomas precession, it is often stated that boosting in $x$, then $y$, then back ...

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group.
Is there a more general theorem that states ...

Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation:
$$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$
However, I ...

I want to understand the relationship of the so common $SU(N)$ and $SO(N)$ groups in physics with the symplectic group which I think is the double cover of the first and the Spin groups $Spin(N)$.
Is ...

In the Virasoro algebra, which is generated by $L_n$, one has the obvious subalgebra spanned by $L_{-1}$ ,$L_{1}$ and $L_{0}$ which is isomorphic to the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$.
The ...

Good evening everybody.
I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's ...

Why does a theory described by a non-abelian group has only a single coupling constant $g$?
While on the other hand in an abelian theory, as Electromagnetism, each charged particle has its own charge ...

On the topic of Pauli matrices, I have noticed that some authors tend to use the term matrix and group interchangeably.
I am asking because I do not see see any profound difference referring to the ...

I'm trying to get a decent understanding of the Bell inequality, and so am trying to understand spin both conceptually and mathematically. When I picture spin, I imagine a sphere rotating about its ...

My question is regarding a vector space and Lie algebra. Why is it that whenever I read advanced physics texts I always hear about Lie algebra? What does it mean to "endow a vector space with a lie ...

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that
$$
3\otimes 3 =6 \oplus \bar{3}~?
$$
I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...